U. Boscain – B. Piccoli
GEOMETRIC CONTROL APPROACH
TO SYNTHESIS THEORY
1. Introduction
In this paper we describe the approach used in geometric control theory to deal with optimiza-tion problems. The concept of synthesis, extensively discussed in [20], appears to be the right mathematical object to describe a solution to general optimization problems for control systems. Geometric control theory proposes a precise procedure to accomplish the difficult task of constructing an optimal synthesis. We illustrate the strength of the method and indicate the weaknesses that limit its range of applicability.
We choose a simple class of optimal control problems for which the theory provides a complete understanding of the corresponding optimal syntheses. This class includes various interesting controlled dynamics appearing in Lagrangian systems of mathematical physics. In this special case the structure of the optimal synthesis is completely described simply by a couple of integers, (cfr. Theorem 3). This obviously provides a very simple classification of optimal syntheses. A more general one, for generic plane control-affine systems, was developed in [18, 10].
First we give a definition of optimal control problem. We discuss the concepts of solution for this problem and compare them. Then we describe the geometric control approach and finally show its strength using examples.
2. Basic definitions
Consider an optimal control problem( )in Bolza form:
˙
x= f(x,u), x∈M,u∈U
min
Z
L(x,u)dt+ϕ(xt erm)
xin=x0, xt erm∈ N⊂M
where M is a manifold, U is a set, f : M×U → T M, L : M ×U → ✁, ϕ : M → ✁, the minimization problem is taken over all admissible trajectory-control pairs(x,u), xinis the initial point and xt erm the terminal point of the trajectory x(·). A solution to the problem( ) can be given by an open loop control u : [0,T ]→U and a corresponding trajectory satisfying the boundary conditions.
One can try to solve the problem via a feedback control, that is finding a function u : M→ U such that the corresponding ODEx˙ = f(x,u(x))admits solutions and the solutions to the Cauchy problem with initial condition x(0)=x0solve the problem( ). Indeed, one explicits the dependence of( )on x0, considers the family of problems =( (x0))x0∈M and tries to
solve them via a unique function u : M → U , that is to solve the family of problems all together.
A well-known approach to the solution of is also given by studying the value function, that is the function V : M→✁assuming at each x0the value of the minimum for the corresponding problem (x0), as solution of the Hamilton-Jacobi-Bellmann equation, see [5, 13]. In general V is only a weak solution to the HJB equation but can be characterized as the unique “viscosity solution” under suitable assumptions.
Finally, one can consider a familyŴof pairs trajectory-control(γx0, ηx0)such that each of
them solves the corresponding problem (x0). This last concept of solution, called synthesis, is the one used in geometric control theory and has the following advantages with respect to the other concepts:
1) Generality
2) Solution description
3) Systematic approach
Let us now describe in details the three items.
1) Each feedback gives rise to at least one synthesis if there are solutions to the Cauchy problems. The converse is not true, that is a synthesis is not necessarily generated by a feedback even if in most examples one is able to reconstruct a piecewise smooth control.
If one is able to define the value function this means that each problem (x0)has a solution and hence there exists at least one admissible pair for each (x0). Obviously, in this case, the converse is true that is every optimal synthesis defines a value function. However, to have a viscosity solution to the HJB equation one has to impose extra conditions.
2) Optimal feedbacks usually lack of regularity and generate too many trajectories some of which can fail to be optimal. See [20] for an explicit example. Thus it is necessary to add some structure to feedbacks to describe a solution. This is exactly what was done in [11, 22].
Given a value function one knows the value of the minimum for each problem (x0). In applications this is not enough, indeed one usually needs to drive the system along the optimal trajectory and hence to reconstruct it from the value function. This is a hard problem, [5]. If one tries to use the concept of viscosity solutions then in the case of Lagrangians having zeroes the solution is not unique. Various interesting problems (see for example [28]) present Lagrangians with zeroes. Recent results to deal with this problem can be found in [16].
3. Optimal synthesis
The approach to construct an optimal synthesis can be summarized in the following way:
a) MP + geometric techniques
⇓
b) Properties of extremal trajectories
⇓
c) Construction of extremal synthesis
⇓ d) Optimal synthesis
We now explain each item of the picture for a complete understanding of the scheme.
a) The Maximum Principle remains the most powerful tool in the study of optimal control problems forty years after its first publication, see [21]. A big effort has been dedicated to generalizations of the MP in recent years, see [6], [23], and references therein.
Since late sixties the study of the Lie algebra naturally associated to the control system has proved to be an efficient tool, see [15]. The recent approach of simplectic geometry proposed by Agrachev and Gamkrelidze, see [1, 4], provides a deep insight of the geometric properties of extremal trajectories, that is trajectories satisfying the Maximum Principle.
b) Making use of the tools described in a) various results were obtained. One of the most famous is the well known Bang-Bang Principle. Some similar results were obtained in [8] for a special class of systems. For some planar systems every optimal trajectory is not bang-bang but still a finite concatenation of special arcs, see [19, 24, 25].
Using the theory of subanalytic sets Sussmann proved a very general results on the regularity for analytic systems, see [26]. The regularity, however, in this case is quite weak and does not permit to drive strong conclusions on optimal trajectories.
Big improvements were recently obtained in the study of Sub-Riemannian metrics, see [2, 3]. In particular it has been showed the link between subanalyticity of the Sub-Riemannian sphere and abnormal extremals.
c) Using the properties of extremal trajectories it is possible in some cases to construct an extremal synthesis. This construction is usually based on a finite dimensional reduction of the problem: from the analysis of b) one proves that all extremal trajectories are finite concatenations of special arcs. Again, for analytic systems, the theory of subanalytic sets was extensively used: [11, 12, 22, 27].
However, even simple optimization problems like the one proposed by Fuller in [14] may fail to admit such a kind of finite dimensional reduction. This phenomenon was extensively studied in [17, 28].
4. Applications to second order equations
and let✁(τ )be the reachable set within timeτfrom the origin. In the framework of [9, 19, 18], we are faced with the problem of reaching from the origin (under generic conditions on F and G) in minimum time every point of✁(τ ). Given a trajectoryγ : [a,b]→✁
2, we define the time alongγ as T(γ )=b−a.
A trajectoryγof (1) istime optimal if, for every trajectoryγ′having the same initial and terminal points, one has T(γ′) ≥ T(γ ). Asynthesis for the control system (1) at timeτ is a family trajectories thenγ1∗γ2denotes their concatenation. For convenience, we define also the vector fields: X=F−G, Y =F+G. We say thatγis an X -trajectory and we writeγ∈Traj(X)if it corresponds to the constant control−1. Similarly we define Y -trajectories. If a trajectoryγis a concatenation of an X -trajectory and a Y -trajectory, then we say thatγis a Y ∗X -trajectory. The time t at which the two trajectories concatenate is called X -Y switching time and we say that the trajectory has a X -Y switching at time t . Similarly we define trajectories of type X∗Y , X∗Y ∗X , etc.
In [19] it was shown that, under generic conditions, the problem of reaching in minimum time every point of the reachable set for the system (1) admits a regular synthesis. Moreover it was shown that✁(τ )can be partitioned in a finite number of embedded submanifolds of dimension 2, 1 and 0 such that the optimal synthesis can be obtained from a feedback u(x)satisfying:
• on the regions of dimension 2, we have u(x)= ±1,
• on the regions of dimension 1, called frame curves (in the following FC), u(x) = ±1 or u(x) = ϕ(x)(whereϕ(x)is a feedback control that depends on F,G and on their Lie bracket [F,G], see [19]). The frame curves that correspond to the feedbackϕare calledturnpikes. A trajectory that corresponds to the control u(t)=ϕ(γ (t))is called a Z -trajectory.
The submanifolds of dimension 0 are called frame points (in the following FP). In [18] it was provided a complete classification of all types of FP and FC.
Given a trajectoryγ ∈ Ŵwe denote by n(γ )the smallest integer such that there existγi ∈
Traj(X)∪Traj(Y)∪Traj(Z), (i=1, . . . ,n(γ )), satisfyingγ =γn(γ )∗ · · · ∗γ1.
The previous program can be used to classify the solutions of the following problem.
Problem: Consider an autonomous ODE in✁:
that describes the motion of a point under the action of a force that depends on the position and on the velocity (for instance due to a magnetic field or a viscous fluid). Then let apply an external force, that we suppose bounded (e.g.|u| ≤1):
¨
y= f(y,y˙)+u. (4)
We want to reach in minimum time a point in the configuration space(y0, v0)from the rest state(0,0).
A deep study of those systems was performed in [9, 10, 19, 18]. From now on we make use of notations introduced in [19]. A key role is played by the functions1A,1B:
1A(x) = det(F(x),G(x)) = x2
The analysis of [19] has to be completed in the following way.
Lemma 4.1 of [19] has to be replaced by the following (see [19] for the definition of Bad(τ )and tanA): local system of coordinates such that G≡(1,0). Then, with the same computations of [19], we have:
Now the proof of Theorem 4.2 of [19] is completed considering the following case:
Note that (4) implies x ∈ tanA. In this case we assume the generic condition ((P1), . . . , (P8) were introduced in [19]):
(P9) 1B(x)6=0.
Suppose X(x)=0 and Y(x)6=0. The opposite case is similar. Choose a new local system of coordinates such that x is the origin, Y =(0,−1)and1−A1(0)= {(x1,x2): x2=0}.
Take U=B(0,r), the ball of radius r centered at 0, and choose r small enough such that:
• 0 is the only bad point in U ;
• 1B(x)6=0 for every x∈U ; • for every x∈U we have:
|X(x)| ≪1. (12)
Let U1 = U∩ {(x1,x2) : x2 > 0}, U2 = U∩ {(x1,x2) : x2 < 0}. We want to prove the following:
THEOREM1. Ifγ ∈ O pt(6)and{γ (t): t∈[b0,b1]} ⊂U then we have a bound on the number of arcs, that is∃Nx ∈ s.t. n( γ|[b0,b1])≤Nx.
In order to prove Theorem 1 we will use the following Lemmas.
LEMMA2. Letγ ∈O pt(6)and assume thatγ has a switching at time t1∈Dom(γ )and that1A(γ (t1))=0. Then1A(γ (t2))=0, t2∈Dom(γ ), iff t2is a switching time forγ.
Proof. The proof is contained in [10].
LEMMA3. Letγ : [a,b] → U be an optimal trajectory such that γ ([a,b]) ⊂ U1or γ ([a,b])⊂U2, then n(γ )≤2.
Proof. It is a consequence of Lemma 3.5 of [19] and of the fact that every point of U1 (respec-tively U2) is an ordinary point i.e.1A(x)·1B(x)6=0.
LEMMA4. Considerγ ∈ O pt(6),{γ (t): t ∈[b0,b1]} ⊂U . Assume that there exist a X -Y switching timet¯∈(b0,b1)forγandγ (t¯)∈U1. Thenγ|[t,b¯ 1]is a Y -trajectory.
Proof. Assume by contradiction thatγ switches at time t′∈(b0,b1), t′>t . If¯ γ (t′)∈U1then this contradicts the conclusion of Lemma 3. Ifγ (t′)∈1−A1(0)then this contradicts Lemma 2. Assumeγ (t′) ∈ U2. From sgn1A(γ (t¯))= −sgn1A(γ (t′))we have that 12X(γ (t¯))∧Y = −12(X(γ (t′))∧Y). This means that:
sgn(X2(γ (t¯))= −sgn(X2(γ (t′)) , (13)
U
γ
x
X Y
X U 1
U 2
t0
t
t’
γ
Figure 1:
of Theorem 1. For sake of simplicity we will writeγinstead ofγ|[b0,b1].
Assume first that no switching happens on1−A1(0). We have the following cases (see fig. 2 for some of these):
(A) γhas no switching; n(γ )=1; (B) for someε >0, γ|[b
0,b0+ε]is an X -trajectory,γ (b0)∈U1, n(γ ) >1;
(B1) ifγswitches to Y in U1, by Lemma 4, n(γ )=2;
(B2) if γ crosses1−A1(0)and switches to Y in U2, by Lemma 3, γ does not switch anymore. Hence n(γ )=2;
(C) for someε >0,γ|[b
0,b0+ε]is an X -trajectory,γ (b0)∈U2, n(γ ) >1;
(C1) ifγswitches to Y before crossing1−A1(0)then, by Lemma 3, n(γ )=2;
(C2) ifγreaches U1without switching, then we are in the (A) or (B) case, thus n(γ )≤2; (D) for someε >0,γ|[b
0,b0+ε]is a Y -trajectory,γ (b0)∈U1, n(γ ) >1;
(D1) ifγswitches to X in U1and never crosses1−A1(0)then by Lemma 3 n(γ )=2;
(D2) ifγ switches to X in U1(at time t0 ∈ [b0,b1]) and then it crosses1−A1(0), then γ|[t
0,b1]satisfies the assumptions of (A) or (C). Hence n(γ )≤3;
(D3) ifγswitches to X in U2at t0∈[b0,b1] and then it does not cross1−A1(0), we have n(γ )=2;
(D4) ifγswitches to X in U2and then it crosses1−A1(0)we are in cases (A) or (B) and n(γ )≤3;
(E) for someε >0, γ|[b
0,b0+ε]is a Y -trajectory,γ (b0)∈U2, n(γ ) >1.
Y
B1 B2
C1 C2
D1 D2 D3 D4 X
Figure 2:
(E2) ifγswitches to X in U2and then it crosses1−A1(0), we are in case (A) or B. Hence n(γ )≤3.
Ifγ switches at1−A1(0), by Lemma 2 all the others switchings ofγhappen on the set1−A1(0). Moreover, ifγ switches to Y it has no more switchings. Hence n(γ )≤3.
The Theorem is proved with Nx =3.
By direct computations it is easy to see that thegeneric conditions P1, . . . ,P9, under which the construction of [19] holds, are satisfied under the condition:
f(x1,0)= ±1 ⇒ ∂1f(x1,0)6=0 (14)
that obviously implies f(x1,0)=1 or f(x1,0)= −1 only in a finite number of points. In the framework of [9, 19, 18] we will prove that, for our problem (5), (6), with the condition (14), the “shape” of the optimal synthesis is that shown in fig. 3. In particular the partition of the reachable set is described by the following
THEOREM2. The optimal synthesis of the control problem (5) (6) with the condition (14), satisfies the following:
1. there are no turnpikes;
2. the trajectoryγ±(starting from the origin and corresponding to constant control±1) exits the origin with tangent vector(0,±1)and, for an interval of time of positive measure, lies in the set{(x1,x2): x1,x2≥0}respectively{(x1,x2): x1,x2≤0};
-1
X1 C
C
+1
+1
+1
-1
-1
Figure 3: The shape of the optimal synthesis for our problem.
point another switching curve generates. The same happens forγ−and the half plane {(x1,x2): x2≤0};
4. let yi, for i = 1, . . . ,n (n possibly+∞) (respectively zi, for i = 1, . . . ,m (m possibly +∞)) be the set of boundary points of the switching curves contained in the half plane {(x1,x2) : x2 ≥ 0} (respectively{(x1,x2) : x2 ≤ 0}) ordered by increasing (resp. decreasing) first components. Under generic assumptions, yiand zi do not accumulate. Moreover:
• For i=2, . . . ,n, the trajectory corresponding to constant control+1 ending at yi starts at zi−1;
• For i =2, . . . ,m, the trajectory corresponding to constant control−1 ending at zi starts at yi−1.
REMARK1. The union ofγ±with the switching curves is a one dimensional 0manifold M. Above this manifold the optimal control is+1 and below is−1.
REMARK2. The optimal trajectories turn clockwise around the origin and switch along the switching part of M. They stop turning after the last yi or zi and tend to infinity with x1(t) monotone after the last switching.
From 4. of Theorem 2 it follows immediately the following:
THEOREM3. To every optimal synthesis for a control problem of the type (5) (6) with the condition (14), it is possible to associate a couple(n,m) ∈ ( ∪ ∞)2 such that one of the following cases occurs:
A. n=m, n finite;
D. n= ∞, m= ∞.
Moreover, ifŴ1, Ŵ2 are two optimal syntheses for two problems of kind (5), (6), (14), and (n1,m1)(resp.(n2,m2)) are the corresponding couples, thenŴ1is equivalent toŴ2iff n1=n2 and m1=m2.
REMARK3. In Theorem 3 the equivalence between optimal syntheses is the one defined in [9].
of Theorem 3. Let us consider the synthesis constructed by the algorithm described in [9]. The stability assumptions (SA1),. . .,(SA6) holds. The optimality follows from Theorem 3.1 of [9].
1. By definition a turnpike is a subset of1−B1(0). From (8) it follows the conclusion.
2. We leave the proof to the reader.
3. Letγ2±(t)= π2(γ±(t)), whereπ2 :✁
2 →
✁,π2(x1,x2)= x2, and consider the adjoint vector fieldv:✁
2×Dom(γ±)×Dom(γ±)→
✁
2associated toγ±that is the solution
of the Cauchy problem:
˙
v(v0,t0;t) = (∇F± ∇G)(γ±(t))·v(v0,t0;t) v(v0,t0;t0) = v0,
(15)
We have the following:
LEMMA5. Consider theγ±trajectories for the control problem (5), (6). We have that v(G,t;0)and G are parallel iff1A(γ±(t))=0 (i.e.γ2±(t)=0).
Proof. Consider the curveγ+, the case ofγ− being similar. From (9) we know that 1A(γ+(t)) = 0 iffγ2+(t) = 0. First assume1A(γ+(t)) = 0. We have that G and (F+G)(γ+(t))are collinear that is G =α(F+G)(γ+(t))withα∈✁. For fixed t0,t the map:
ft0,t:v07→v(v0,t0;t)
(16)
is clearly linear and injective, then using (15) andγ˙+(t)=(F+G)(γ+(t)), we obtain v(G,t;0)=αv (F+G)(γ+(t)),t;0
=α(F+G)(0)=αG.
Viceversa assumev(G,t;0) = αG, then (as above) we obtainv(G,t;0) = αv((F + G)(γ+(t)),t;0). From the linearity and the injectivity of (16) we have G = α(F + G)(γ+(t))hence1A(γ+(t))=0.
LEMMA6. Consider the trajectoryγ+for the control problem (5), (6). Lett¯>0 (pos-sibly+∞) be the first time such thatγ2+(t¯)=0. Thenγ+is extremal exactly up to time ¯
t. And similarly forγ−.
Proof. In [19] it was defined the functionθ (t)= arg G(0), v G(γ+(t)),t,0
. This function has the following properties:
(ii) γ+is extremal exactly up to the time in which the measure of the range ofθisπi.e. up to the time:
t+=min{t∈[0,∞] :|θ (s1)−θ (s2)| =π, for some s1,s2∈[0,t+]}, (17)
under the hypothesisθ (˙ t+)6=0. This was proved in Proposition 3.1 of [9]. From Lemma 5 we have that1A(γ+(t))=0 iff there exists n∈ satisfying:
θ (G, v(G,t,0))=nπ . (18)
In particular (18) holds for t= ¯t and some n. From the fact thatt is the first time in which¯
γ2+(t¯)=0 and hence the first time in which1A(γ+(t¯))=0, we have that n=1.
Fromθ (t¯)=πand sgn(θ (˙ t¯))=1 the Theorem is proved with t+= ¯t.
From Lemma 6,γ±are extremal up to the first intersection with the x1-axis.
Let t be the time such thatγ−(t)=z1, defined in 4 of Theorem 2. The extremal trajecto-ries that switch along the C-curve starting at y1(if it exists), are the trajectories that start from the origin with control−1 and then, at some time t′<t, switch to control+1. Since the first switching occurs in the orthant{(x1,x2): x1,x2<0}, by a similar argument to the one of Lemma 6, the second switch has to occur in the half space{(x1,x2): x2>0}, because otherwise between the two switches we have meas(range(θ (t))) > π. This proves that the switching curves never cross the x1-axis.
4. The two assertions can be proved separately. Let us demonstrate only the first, being the proof of the second similar. Define y0=z0=(0,0). By definition the+1 trajectory starting at z0reaches y1. By Lemma 2 we know that if an extremal trajectory has a switching at a point of the x1-axis, then it switches iff it intersects the x1-axis again. This means that the extremal trajectory that switches at yihas a switching at zj for some j . By induction one has j≥i−1. Let us prove that j =i−1. By contradiction assume that j>i−1, then there exists an extremal trajectory switching at zi−1that switches on the C curve with boundary points yi−1, yi. This is forbidden by Lemma 2.
EXAMPLES1. In the following we will show the qualitative shape of the synthesis of some physical systems coupled with a control. More precisely we want to determine the value of the couple(m,n)of Theorem 3.
Duffin Equation
The Duffin equation is given by the formulay¨ = −y−ε(y3+2µy˙),ε, µ > 0,εsmall. By introducing a control term and transforming the second order equation in a first order system, we have:
˙
x1 = x2 (19)
˙
x2 = −x1−ε(x31+2µx2)+u. (20)
From this form it is clear that f(x)= −x1−ε(x13+2µx2).
+1 X2
X1 C
C
-1
Figure 4: The synthesis for the Van Der Pol equation.
moves in the orthant:= {(x1,x2), x1,x2>0}. To know the shape of the synthesis we need to know where(F+G)2(x)=0. If we set a= 2εµ1 , this happens where
x2=a(1−x1−εx3) . (21)
From (19) and (20) we see that, after meeting this curve, the trajectory moves withγ˙1+>0 and
˙
γ2+ <0. Then it meets the x1-axis because otherwise ifγ+(t)∈ we necessarily have (for t→ ∞)γ1+→ ∞,γ˙2+→0, that is not permitted by (20). The behavior of the trajectoryγ−is similar.
In this case, the numbers(n,m) are clearly(∞,∞)because the +1 trajectory that starts at z1meets the curve (21) exactly one time and behaves likeγ+. So the C-curve that starts at y1 meets again the x1axis. The same happens for the−1 curve that starts at y1. In this way an infinite sequence of yi and zi is generated.
Van der Pol equation
The Van der Pol equation is given by the formulay¨= −y+ǫ(1−y2)y˙+u,ε >0 and small. The associated control system is:x˙1=x2,x˙2= −x1+ε(1−x21)x2+u. We have(F+G)2(x)=0 on the curves x2= −ε(x11+1) for x16= ±1, x1= −1. After meeting these curves, theγ
+
tra-jectory moves withγ˙1+>0 andγ˙2+<0 and, for the same reason as before, meets the x1-axis. Similarly forγ−. As in the Duffin equation, we have that m and n are equal to+∞. But here, starting from the origin, we cannot reach the regions: n(x1,x2): x1<−1,x2≥ −ε(x1
1+1)
o
,
n
(x1,x2): x1>−1,x2≤ −ε(x11−1)
o
(see fig. 4).
Another example
equa-A X2
X1
C
+1
+1
C
B
-1
Figure 5: The synthesis for the control problem (22), (23). The sketched region is reached by curves that start from the origin with control−1 and then switch to +1 control between the points A and B.
tion:y¨= −ey+ ˙y+1. The associated control system is:
˙
x1 = x2 (22)
˙
x2 = −ex1+x2+1+u (23)
We haveγ˙2+=0 on the curve x2=ex1−2. After meeting this curve, theγ+trajectory meets the x1-axis.
Now the synthesis has a different shape because the trajectories corresponding to control−1 satisfyγ˙2=0 on the curve:
x2=ex1 (24)
that is contained in the half plane{(x1,x2): x2>0}. Henceγ−never meets the curve given by (24) and this means that m=0. Since we know that n is at least 1, by Remark 4, we have n=1, m=0. The synthesis is drawn in fig. 5.
5. Optimal syntheses for Bolza Problems
Quite easily we can adapt the previous program to obtain information about the optimal synthe-ses associated (in the previous sense) to second order differential equations, but for more general minimizing problems.
We have the well known:
LEMMA7. Consider the control system:
˙
x=F(x)+uG(x), x∈✁
2, F,G∈ 3(
✁
2,
✁
Let L :✁
2→
✁ be a
3bounded function such that there existsδ >0 satisfying L(x) > δfor any x∈✁
2.
Then, for every x0∈✁
2, the problem: minRτ
0 L(x(t))dt s.t. x(0)=0, x(τ )=x0, is equivalent to the minimum time problem (with the same boundary conditions) for the control systemx˙ = F(x)/L(x)+uG(x)/L(x).
By this lemma it is clear that if we have a second order differential equation with a bounded-external forcey¨= f(y,y˙)+u, f ∈ 3(✁
2), f(0,0)=0,|u| ≤1, then the problem of reaching a point in the configuration space(y0, v0)from the origin, minimizing
Rτ
0 L(y(t),y˙(t))dt , (under the hypotheses of Lemma 7) is equivalent to the minimum time problem for the system: x˙1 = x2/L(x), x˙2 = f(x)/L(x)+1/L(x)u. By setting: α : ✁
The equations defining turnpikes are:1A 6=0,1B =0, that with our expressions become the differential conditionα+x2∂2α=0 that in terms of L is:
L(x)−x2∂2L(x)=0 (26)
REMARK4. Since L > 0 it follows that the turnpikes never intersect the x1-axis. Since (26) depends on L(x)and not on the control system, all the properties of the turnpikes depend only on the Lagrangian.
Now we consider some particular cases of Lagrangians.
L=L(y) In this case the Lagrangian depends only on the position y and not on the velocityy˙
(i.e. L=L(x1)). (26) is never satisfied so there are no turnpikes.
L=L(y˙) In this case the Lagrangian depends only on velocity and the turnpikes are horizontal lines.
L=V(y)+12y˙2 In this case we want to minimize an energy with a kinetic part12y˙2and a pos-itive potential depending only on the position and satisfying V(y) >0. The equation for turnpikes is(x2)2=2V(x1).
References
[1] AGRACHEVA., Methods of Control Theory, in “Mathematic Geometry”, Proc. ICM 94, Dekker, New York 1998.
[2] AGRACHEVA., BONNARDB., CHYBAM., KUPKAI., Sub Riemannian Sphere in Mar-tinet Flat Case, ESAIM: COCV 2 (1997), 337–448.
[3] AGRACHEVA., ELALAOUIC., GAUTHIERJ. P., Sub Riemannian Metrics on✁
3, to ap-pear in Geometric Control and Non-Holonomic Problems in Mechanics, Conference Pro-ceeding Series, Canad. Math Soc.
[4] AGRACHEVA., GAMKRELIDZER., Simplectic Methods for Optimization and Control, in “Geometry of Feedback and Optimal Control”, edited by B. Jakubczyk and W. Respondek, Dekker, New York 1998.
[6] BIANCHINIR. M., STEFANIG., Graded Approximations and Controllability along a Tra-jectory, SIAM J. Control and Optimization 28 (1990), 903–924.
[7] BOLTIANSKII, Sufficient Conditions for Optimality and the Justification of the Dynamic Programming Principle, SIAM J. Control and Optimization 4 (1966), 326–361.
[8] BRESSANA., PICCOLIB., A Baire Category Approach to the Bang-Bang Property, Jour-nal of Differential Equations 116 (2) (1995), 318–337.
[9] BRESSANA., PICCOLIB., Structural Stability for Time-Optimal Planar Syntheses, Dy-namics of Continuous, Discrete and Impulsive Systems 3 (1997), 335-371.
[10] BRESSANA., PICCOLIB., A Generic Classification of Time Optimal Planar Stabilizing Feedbacks, SIAM J. Control and Optimization 36 (1) (1998), 12–32.
[11] BRUNOVSKIP., Every Normal Linear System Has a Regular Time-Optimal Synthesis, Math. Slovaca 28 (1978), 81–100.
[12] BRUNOVSKYP., Existence of Regular Syntheses for General Problems, J. Diff. Equations 38 (1980), 317–343.
[13] FLEMING W. H., SONER M., Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York 1993.
[14] FULLERA. T., Relay Control Systems Optimized for Various Performance Criteria, Auto-matic and remote control, Proc. first world congress IFAC Moscow 1, Butterworths 1961, 510–519.
[15] JURDJEVICV., Geometric Control Theory, Cambridge University Press 1997.
[16] MALISOFFM., On The Bellmann Equation for Control Problem with Exit Time and Un-bounded Cost Functionals, to appear on Proc. of 39th CDC of IEEE.
[17] MARCHALC., Chattering arcs and chattering controls, J. Optim Theory Appl. 11 (1973), 441–468.
[18] PICCOLIB., Classifications of Generic Singularities for the Planar Time-Optimal Synthe-sis, SIAM J. Control and Optimization 34 (6), December 1996, 1914–1946.
[19] PICCOLIB., Regular Time-Optimal Syntheses for Smooth Planar Systems, Rend. Sem Mat. Univ. Padova 95 (1996), 59–79.
[20] PICCOLIB., SUSSMANNH. J., Regular Synthesis and Sufficiency Conditions for Optimal-ity, to appear on SIAM J. Control and Optimization.
[21] PONTRYAGINL. S., BOLTIANSKIV., GAMKRELIDZER., MICHTCHENKOE.,The Math-ematical Theory of Optimal Processes, John Wiley and Sons, Inc, 1961.
[22] SUSSMANNH. J., Subanalytic sets and feedback control, J. Diff. Equations 31 (1) (1979), 31–52.
[23] SUSSMANN H. J., Geometry and Optimal Control in “Mathematical Control Theory”, Springer-Verlag, New York 1998, 140–198.
[24] SUSSMANNH. J., The Structure of Time-Optimal Trajectories for Single-Input Systems in the Plane: the C∞Nonsingular Case, SIAM J. Control Optim. 25 (2) (1987), 433–465.
[25] SUSSMANNH. J., The Structure of Time-Optimal Trajectories for Single-Input Systems in the Plane: the General Real-Analytic Case, SIAM J. Control Optim. 25 (4) (1987), 868–904.
[27] SUSSMANN H. J., Regular Synthesis for Time-Optimal Control of Single-Input Real-Analytic Systems in the Plane, SIAM J. Control Optim. 25 (5) (1987), 1145–1162.
[28] ZELKINM. I., BORISOVV. F., Theory of Chattering Control, with Applications to Astro-nautics, Robotics, Economics and Engineering, Birkh¨auser, Boston 1994.
AMS Subject Classification: ???.
Benedetto PICCOLI
DIIMA, Universit`a di Salerno Via Ponte Don Melillo 84084 Fisciano (SA), ITALY and
SISSA–ISAS Via Beirut 2-4 34014 Trieste, ITALY
